Asserting something one does not believe is lying. By the principle of charity we should assume our fellows are not lying, in which case "X" implies "I believe X". Obviously, that's only halfway to equivalence.

If I were to say, "I believe that the president is John McCain", and you responded by disputing my claim that the president is John McCain, I would be out of line to respond that I had never asserted that the president is John McCain. Similarly for the exchange "I believe that Annoyance is Caledonian" "But I'm not Caledonian" "I didn't say you were".

And so they are equivalent, unless you deny the principle of charity or have a counterexample for my second point.